Strong covering without squares

The study of “covering lemmas” started with Jensen [DeJe] who proved in 1974–5 that in the absence of 0 there is a certain degree of resemblance between V and L. More precisely, if 0 does not exist then for every set of ordinals X there exists a set of ordinals Y ∈ L such that X ⊆ Y and V 2 |Y | = max{|X|,א1}. There is no hope of covering countable sets by countable ones in general, because doing Namba forcing over L will change the cofinality of א2 to ω while preserving א1. This form of covering has strong implications for the structure of V . For example Jensen’s theorem implies that in the absence of 0 the Singular Cardinals Hypothesis holds, and that there is a special κ+-Aronszajn tree for every singular κ. So we can conclude that the negations of these statements have substantial consistency strength. One subsequent line of development has involved proving covering lemmas over larger and larger “core models”, on the assumption of the nonexistence of stronger and stronger large cardinals. Inevitably these covering lemmas have much more complex statements than Jensen’s original theorem, the reason being that once the core model contains even one measurable cardinal we can start to do Prikry forcing.